Timeline for Under exactly what (extra) conditions (if any) is a connected Hausdorff manifold with a Riemannian metric a metric space?
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| May 29, 2013 at 7:39 | comment | added | Benjamin Dickman | An answer to my question (see comment above) posed last August: mathoverflow.net/questions/104965/… | |
| May 19, 2013 at 17:54 | answer | added | Peter Michor | timeline score: 11 | |
| Aug 19, 2012 at 21:09 | comment | added | Jeff Rubin | I also found a note in Abraham and Marsden, "Foundations of Mechanics", Updated 1985 Printing, p128: "Recall that we include second countable in our definition of a manifold. It is interesting that a manifold which admits a Riemannian metric (or a connection) must be second countable (see Abraham [1963])." Unfortunately, the only entry in the references that could possibly match that is Abraham, R. 1963.a Transversality in manifolds of mappings. Bull. Am. Math. Soc. 69 (4):470-474. and that has nothing to do with Riemannian metrics or second countability. | |
| Aug 16, 2012 at 22:43 | comment | added | Benjamin Dickman | Is there a Hausdorff Banach manifold which is not a regular space? maik.ru/full/rusmath/97/10/rusmath10_97p49full.pdf nonchalantly states (p.53): "Note that a Hausdorff Banach manifold X is a regular space." Separately, in 1.3.1 of "Momentum Maps and Hamiltonian Reduction" (tinyurl.com/budt9hs): "it can proved that a connected Hausdorff manifold admits a Riemannian metric if and only if it is second countable." Unfortunately, neither note is sourced; otherwise you'd have regularity and second-countability, and Urysohn would finish things off without any extra conditions. | |
| Aug 12, 2012 at 19:54 | history | asked | Jeff Rubin | CC BY-SA 3.0 |